Since the KdV equation has derivatives higher than two, a dynamic interpretation
of the same is not possible at the level of the equation itself. In order to employ the quasi-integrability mechanism of
Ref.s [1]}, [2]}, [3]}, [4]}, the notion of potential is essential, that emerges from such an interpretation of the equation.
In case of KdV system, however, a well-known Hamiltonian formulation [5]} exists. In fact, the KdV equation REF 
can be shown to emerge from two different equivalent Hamiltonians. Subjected to the order of non-linearity appearing in
the Lax pair of Eq. REF , we opt for the following Hamiltonian,
\(H_1[u]=\int _{-\infty }^\infty dx~\left(\frac{1}{2}u_x^2-u^3\right) \quad {\rm with}\quad \frac{\delta H_1[u]}{\delta u(x)}=-3u^2-u_{xx}.\) 
